Size-Dependent Effects of Micro-Nano Mindlin Plates Based on the Couple Stress Theory
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摘要:
基于偶应力理论，建立了适用于微纳米结构的Mindlin板理论。考虑横向剪切变形和材料的尺度效应并引入长度尺寸参数，推导了各向同性微纳米Mindlin板的本构方程。根据板的平衡条件，进一步推导出用位移函数和转角函数表示的板的屈曲和振动控制方程。通过对位移和转角变量进行空间和时间域上的分离，得出了四边简支（SSSS）和对边简支、对边固支（SCSC）两种边界情况下微纳米板的屈曲和振动问题的解析解。然后利用MATLAB软件进行算例分析，获得了不同尺寸参数、长宽比、厚长比等情况下板的临界屈曲荷载和固有频率。研究结果与已有文献中的结果以及ABAQUS有限元仿真解进行对比，结果表明，不同参数下的三种方法得到的结果均十分接近。算例分析发现，尺度效应对屈曲载荷和固有频率都有显著影响。
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关键词:
- 偶应力 /
- 尺度效应 /
- Mindlin板理论 /
- 屈曲 /
- 振动
Abstract:A Mindlin plate theory for micro-nano structures was proposed based on the couple stress theory. A length parameter was introduced to consider the size effect, and the constitutive equations for the micro-nano Mindlin plate were derived in view of the transverse shear deformation. The buckling and free vibration governing equations in terms of displacements and the slope functions of the shear deformation micro-nano plate were further deduced with the force equilibrium conditions. The analytical solutions of buckling and free vibration for the shear deformation micro-nano plate were obtained through separation of the displacement and rotation variables in space and time domains. Two scenarios of boundary conditions were analyzed: SSSS (simply supported by 4 edges) and SCSC (2 opposite edges simply supported and other 2 edges clamped). A MATLAB program was developed to compute the critical buckling and natural frequencies with different values of dimensional parameters, aspect ratios and length-to-thickness ratios. The research results, in comparison with those from the ABAQUS finite element analysis and previous literatures, are consistent with the latter ones. The examples show that, the size effects significantly influence the buckling load and the natural frequency.
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Key words:
- couple stress /
- size effect /
- Mindlin plate theory /
- buckling /
- vibration
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图 2 微元体各面上的应力和偶应力分布状态
Figure 2. The distributions of the stress and the couple stress on the surfaces of a cubic micro element
图 3 各向同性微纳米板微元体的受力平衡状态
Figure 3. The force and moment equilibrium of a micro element of the isotropic micro-nano plate
图 6 SSSS和SCSC边界下微纳米板的临界屈曲荷载随厚长比的变化
Figure 6. Effects of the relative depth on the buckling load of the micro-nano plate under SSSS and SCSC
图 7 不同厚长比下，微纳米板临界屈曲荷载随尺寸参数的变化：（a）SSSS；（b）SCSC
Figure 7. Effects of the dimensional parameters on the buckling load of the micro-nano plate for different thickness-to-length ratios: (a) SSSS; (b) SCSC
图 8 不同尺寸参数下，SSSS微纳米板固有频率随厚长比的变化
Figure 8. Effects of the thickness-to-length ratio on the natural frequency of the SSSS micro-nano plate for different values of dimensional parameters
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